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| In [[fluid dynamics]], '''Sauter mean diameter''' ('''SMD''', ''d''<sub>32</sub> or ''D''[3, 2]) is an average of particle size. It was originally developed by German scientist J. Sauter in the late 1920s.<ref>Sauter J. "Die Grössenbestimmung der in Gemischnebeln von Verbrennungskraftmaschinen vorhandenen Brennstoffteilchen" VDI-Forschungsheft Nr. 279 (1926) und Nr. 312 (1928)</ref> It is defined as the [[diameter]] of a sphere that has the same [[volume]]/[[surface area]] ratio as a particle of interest.
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| Several methods have been devised to obtain a good estimate of the SMD.
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| SMD is typically defined in terms of the surface diameter, ''d''<sub>''s''</sub>:
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| :<math>d_s = \sqrt{\frac{A_p}{\pi}}</math>
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| and volume diameter, ''d''<sub>''v''</sub>:
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| :<math>d_v = \left(\frac{6 V_p}{\pi}\right)^{1/3},</math>
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| where ''A''<sub>''p''</sub> and ''V''<sub>''p''</sub> are the surface area and volume of the particle, respectively. If ''d''<sub>''s''</sub> and ''d''<sub>''v''</sub> are measured directly by other means without knowledge of ''A''<sub>''p''</sub> or ''V''<sub>''p''</sub>, Sauter diameter for a given particle is
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| :<math>SD = D[3,2] = d_{32} = \frac{d_v^3}{d_s^2}. </math>
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| If the actual surface area, ''A''<sub>''p''</sub> and volume, ''V''<sub>''p''</sub> of the particle are known the equation simplifies further:
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| :<math>\frac{V_p}{A_p} = \frac{\frac{4}{3}\pi (d_v/2)^3}{4\pi (d_s/2)^2} =
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| \frac{(d_v/2)^3}{3 (d_s/2)^2} = \frac{d_{32}}{6}</math>
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| :<math>d_{32} = 6\frac{V_p}{A_p}.</math>
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| This is usually taken as the [[mean]] of several measurements, to obtain the Sauter <u>mean</u> diameter, SMD:
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| <!--
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| Formula looks wrong. See:
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| * http://www.silver-colloids.com/Tutorials/psintro.html
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| * http://me.queensu.ca/people/sellens/research/sprayFlow/preusser/diplomar/node13.html
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| :<math>SMD = \bar{d_{32}} = \sum_i\frac{d_v^3}{d_s^2}</math>
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| -->
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| This provides intrinsic data that help determine the particle size for fluid problems.
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| ==Applications==
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| The SMD can be defined as the [[diameter]] of a drop having the same [[volume]]/[[surface area]] ratio as the entire spray.
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| :<math>D_s = \frac{1}{\sum_i \frac{f_i}{d_i}}</math>
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| ::<math>f_i</math> is the [[scalar (mathematics)|scalar]] variable for the dispersed phase
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| ::<math>d_i</math> is the discrete bubble size | |
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| SMD is especially important in calculations where the active surface area is important. Such areas include catalysis and applications in fuel combustion.
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| ==See also==
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| *[[Sphericity]]
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Sauter Mean Diameter}}
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| [[Category:Fluid dynamics]]
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